Chapter 11 · Question 5

On what factors does the resistance of a conductor depend? Write the formula relating resistance and resistivity, and define resistivity.

Answer Revealed

Direct Answer:

Resistance of a conductor depends on (i) its length

l

(

R \propto l

), (ii) its area of cross-section

A

(

R \propto 1/A

), and (iii) the nature of the material (resistivity

\rho

). The formula is

R = \rho l / A

. Resistivity is the resistance offered by a conductor of unit length and unit cross-sectional area at a given temperature.

Simple Explanation

Resistance depends on three things: how long the conductor is (longer = more resistance), how thick it is (thicker = less resistance, like a wider pipe lets more water flow), and what it is made of (copper resists less than nichrome). The formula

R = \rho l / A

puts all three together. Resistivity (

\rho

) is just a number that tells you how strongly a given material opposes current.

Exam-Ready Structure

The resistance of a conductor depends on three main factors, and resistivity ties them together in a single material property: 1. Effect of length (

l

): Resistance is directly proportional to the length of the conductor (

R \propto l

). If the length doubles, resistance also doubles, assuming area and material remain the same. 2. Effect of area of cross-section (

A

): Resistance is inversely proportional to the area of cross-section of the conductor (

R \propto 1/A

). A thicker wire has more cross-sectional area and therefore less resistance. 3. Effect of nature of material: Different materials have different abilities to resist current flow. This is captured by resistivity (

\rho

). 4. Formula: Combining these factors gives

R = \rho l / A

, where

R

is resistance (in ohms),

\rho

is resistivity of the material (in ohm-metre,

\Omega\ \text{m}

l

is length (in m), and

A

is cross-sectional area (in

\text{m}^2

). 5. Definition of resistivity: The resistivity of a substance is numerically equal to the resistance of a conductor of that substance having unit length (1 m) and unit cross-sectional area (1

\text{m}^2

). It is a characteristic property of the material — copper has low resistivity and is a good conductor, while nichrome has high resistivity and is used in heating elements. 6. Alloys like nichrome have higher resistivity than their constituent metals, which is why they are preferred in heating devices.

Key Points

Resistance depends on length ( $R \propto l$ ), area of cross-section ( $R \propto 1/A$ ), and material
Formula: $R = \rho l / A$
Resistivity ( $\rho$ ) is the resistance of a unit cube of the material ( $1\ \text{m}$ length, $1\ \text{m}^2$ area)
SI unit of resistivity is ohm-metre ( $\Omega\ \text{m}$ )
Alloys have higher resistivity than pure metals and are used in heating elements

Common Mistakes

Saying resistance is directly proportional to area (it is inversely proportional — thicker wire has less resistance)
Confusing resistivity with conductivity (resistivity is the inverse property)

On what factors does the resistance of a conductor depend? Write the formula relating resistance and resistivity, and define resistivity.